Definition of Power Set

A set is a collection of different objects having some common property. These objects are the elements of the set. The elements have their own identity separately so all the elements make the subsets of the set.

And the set of all the subsets is the power set of that particular set, including null set and itself also.

If A is the set then the set of all the subsets of A is the power set of A. The power set of A will be represented by P(A).

The power set of Z is the set of all the subsets of Z.We will list all the subsets then enclose them in the curly braces “{ }”

P(Z) = { { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {1, 5}, {1, 3, 5} }

Example

Here the two elements of the set are apple and banana.

So the subsets of it could be empty set, apple, banana,apple and banana both.

The power set of this set will be the set of all the subsets as shown above in picture.

Cardinality of Power Set

Cardinality is the number of elements of a set.The number of elements of a power set is the number of subsets.The number of elements is represented by |Z|. As we know that the formula of calculating number of subsets is 2n so the number of elements of a power set will also be 2n as the power set is the set of all the subsets of a set.

Power Set of Empty Set

The empty set is the set having no or zero element. Its cardinality is zero(0).It is also called null or void set.

The power set is the set of all the subsets of a set. We know that every set is a subset of itself and empty set is also the subset of itself.

So the set containing only the empty set is the power set of an empty set.

If A is the empty set i.e. A = ∅

P(A) = {∅}

This is a singleton set as the set having only one element is called the singleton set.

The important point is that the empty set is the set with zero element but the power set of an empty set is not empty set because the power set of an empty set contains one element that is ∅.

What is the Power Set of a set having Empty Set as an element?

First,we need to understand the difference between the empty set and the set with the empty set as an element. In the empty set, we don’t have any element i.e. the zero cardinality but if we have a set with “∅” as the element, then this shows that it has one element.

Example

Z= {}, i.e. Z is the empty set having zero element.

Y= {∅}, i.e. Y is a set with one element i.e. ∅.

Here the power set of Z i.e. empty set is {∅}

P (Z) = {∅}

But the power set of Y=21=2

Subsets of Y are ∅ as empty set is subset of every set and {∅} as the element of set Y.So the power set of Y is

P(Y) = {∅,{∅} }

Power Set of a Power Set

Aswe know that the power set is the set of all the subsets of any given set then what about the power set of a power set?Let’s understand it with an example-

M = {a}, |M|=1, i.e. the number of elements of M is 1. So the number of elements of power set of M will be 21= 2.

P(M) = { { },{a} }

The subsets of P(M) are { },

{{ }},

{{a}},

{ { },{a}}

|P(P(M))| = 22 = 4

P(P(M)) = { { }, {{ }}, {{a}}, { { }, {a}}}

And if we want to calculate the power set of the power set of the power set of M i.e. P(P(P(M))), again then we can do it with the same process again as done above.

Power Sets and Partially ordered Sets

All the elements of a powerset have some natural ordering, ⊆, and any power set and ⊆ form a set ordered. A set is an ordered set if their elements have an order, and a partially ordered set is said so because it’s all pairs of elements are not ordered.